List of top Mathematics Questions asked in VITEEE

For which of the following values of \( m \), the area of the region bounded by the curve \( y = x - x^2 \) and the line \( y = mx \) equals 5?

The domain of the function \[ f(x) = \frac{1}{\log(1 - x)} + \sqrt{x + 2} \] is

If \( f(x) = \left\{ \begin{array}{ll} 1 + \left| \sin x \right|, & \text{for } -\pi \leq x<0
e^{x/2}, & \text{for } 0 \leq x<\pi
\end{array} \right. \) then the value of \( a \) and \( b \), if \( f \) is continuous at \( x = 0 \), are respectively

If \( n = 1999 \), then \( \sum_{i=1}^{1999} \log x_i \) is equal to

In the expansion of \( a + bx \), the coefficient of \( x^r \) is

\( P \) is a fixed point \( (a, a, a) \) on a line through the origin equally inclined to the axes, then any plane through \( P \) perpendicular to \( OP \), makes intercepts on the axes, the sum of whose reciprocals is equal to

If \( R \to R \) be such that \( f(1) = 3 \) and \( f'(1) = 6 \), then \( f(x) \) is equal to

The locus of mid points of tangents intercepted between the axes of ellipse \[ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \] is

If a tangent having slope \( \frac{-4}{3} \) to the ellipse \[ \frac{x^2}{18} + \frac{y^2}{32} = 1 \] intersects the major and minor axes in points A and B respectively, then the area of \( \triangle OAB \) is equal to

The sides \( AB \), \( BC \), and \( CA \) of a triangle \( \triangle ABC \) have respectively 3, 4, and 5 points lying on them. The number of triangles that can be constructed using these points as vertices is

The value of the integral \[ \int_1^\infty \frac{x+1}{|x-1|} \left( \frac{x-1}{x+1} \right)^{1/2} \, dx \] is

If \( f(x) = x - \lfloor x \rfloor \), for every real number \( x \), where \( \lfloor x \rfloor \) is the integral part of \( x \), then \[ \int f(x) \, dx \] is equal to

\( \int (x + 1)(x - x^2) e^x \, dx \) is equal to

If \( P \) is a double ordinate of hyperbola \[ \frac{x^2}{a^2} - \frac{y^2}{b^2} = 1 \] such that \( OPQ \) is an equilateral triangle, \( O \) being the centre of the hyperbola, then the eccentricity \( e \) of the hyperbola satisfies

The number of integral values of \( K \), for which the equation \( 7 \cos x + 5 \sin x = 2K + 1 \) has a solution, is

The value of the determinant \[ \begin{vmatrix} \cos(\alpha - \beta) & \cos \alpha & \cos \beta
\cos(\alpha - \beta) & 1 & \cos \beta
\cos \alpha & \cos \beta & 1 \end{vmatrix} \]

If the points \( (x_1, y_1) \), \( (x_2, y_2) \), and \( (x_3, y_3) \) are collinear, then the rank of the matrix \[ \begin{bmatrix} x_1 & y_1 & 1
x_2 & y_2 & 1
x_3 & y_3 & 1 \end{bmatrix} \]

The line joining two points \( A(2,0) \), \( B(3,1) \) is rotated about \( A \) in anti-clockwise direction through an angle of \( 15^\circ \). The equation of the line in the new position is

The line \( 2x + \sqrt{6}y = 2 \) is tangent to the curve \( x^2 - 2y^2 = 4 \). The point of contact is

The number of integral points (integral point means both the coordinates should be integers) exactly in the interior of the triangle with vertices \( (0, 0), (0, 21), (21, 0) \) is

The vector \( \mathbf{r} = 3\hat{i} + 4\hat{k} \) can be written as the sum of a vector \( \mathbf{v} \), parallel to \( \hat{i} + \hat{k} \), and a vector \( \mathbf{u} \), perpendicular to \( \hat{i} + \hat{k} \). Then, the value of \( \mathbf{v} \) is

The centres of a set of circles, each of radius 3, lie on the circle \( x^2 + y^2 = 25 \). The locus of any point in the set is

The angle of intersection of the circles \( x^2 + y^2 - 8x - 9 = 0 \) and \( x^2 + y^2 + 2x - 4y - 11 = 0 \) is

If \( x^2 + 2x - 5 = 0 \), then the values of \( x \) are

\( \theta \) and \( \gamma \) are the roots of the equation \( x^2 - \alpha x + \beta = 0 \) and if \( \theta + \gamma = \alpha \), then what is the value of \( \theta^2 + \gamma^2 \)?